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THESIS

A 3D SPATIAL CHANNEL MODEL

FOR CELLULAR RADIO

by

Christos Sasiakos

September 2000

Thesis Advisor:

Ramakrishna Janaswamy

Second Reader:

Tri T. Ha

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Master's Thesis

4. TITLE AND SUBTITLE: A 3D Spatial Channel model for cellular radio.

6. AUTHOR(S) Christos Sasiakos

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Naval Postgraduate School

Monterey, CA 93943-5000

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13. ABSTRACT (maximum 200 words)

This thesis provides closed form expressions for the angular distribution in azimuth and elevation planes for a geometrically
based single bounce spheroid model. The geometry of the spheroid is defined by the semi-major axis a and the semi-minor axis
b. The other parameter of interest in the model is the distance D between the base station and the mobile station. The latter is
assumed to be at the center of the spheroid. The mobile station is assumed to be the transmitter, while the base station is the
receiver. This thesis investigates the effects of the above parameters on the angular distribution of the received waves.
Important parameters such as the r.m.s angle spread in azimuth and elevation plane are calculated from the p.d.f. expressions
derived. The behaviour of these r.m.s angle spreads versus the ratio a/D or b/D respectively is also investigated.

14. SUBJECT TERMS Spatial channel Model, Joint TOA/AOA pdf.
Azimuth Plane. AOA Marginal pdf in Elevation Plane, r.m.s angle spread

AOA Marginal pdf in

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Approved for public release; distribution is unlimited

A 3D SPATIAL CHANNEL MODEL
FOR CELLULAR RADIO

Christos Sasiakos

Lieutenant, Hellenic Navy

B.S.E.E. , Hellenic Naval Academy, 1988

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

Septemtfepzt

DUDLEY KNOX LIBRARY

NAVAL POSTGRADUATE SCHOOL

MONTEREY CA 93943-5101

ABSTRACT

This thesis provides closed form expressions for the angular distribution in
azimuth and elevation planes for a geometrically based single bounce spheroid model.
The geometry of the spheroid is defined by the semi-major axis a and the semi-minor
axis b. The other parameter of interest in the model is the distance D between the base
station and the mobile station. The latter is assumed to be at the center of the spheroid.
The mobile station is assumed to be the transmitter, while the base station is the receiver.
This thesis investigates the effects of the above parameters on the angular distribution of
the received waves. Important parameters such as the r.m.s angle spread in azimuth and
elevation plane are calculated from the p.d.f. expressions derived. The behaviour of these
r.m.s angle spreads versus the ratio a/D or b/D respectively is also investigated.

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VI

TABLE OF CONTENTS

I. INTRODUCTION 1

A. BACKGROUND 1

B. OBJECTIVE 9

II. PROBLEM FORMULATION 11

A. INTRODUCTION 11

B. DERIVATION OF THE JOINT TOA/AOA P.D.F.S 13

C. DERIVATION OF MARGINAL AOA PROBABILITY DENSITY
FUNCTION 17

D. MARGINAL AOA P.D.F IN AZIMUTH PLANE FOR THE
SPHEROID SPATIAL CHANNEL MODEL 19

E. MARGINAL AOA P.D.F IN AZIMUTH PLANE FOR THE
SPECIAL CASE OF THE SEMI-MINOR AXIS OF THE
SPHEROID GOING ZERO 25

F. DERIVATION OF MARGINAL AOA P.D.F IN AZIMUTH PLANE
THROUGH A SECOND APPROACH 26

G. MARGINAL AOA P.D.F IN ELEVATION PLANE FOR THE
SPHEROID SPATIAL CHANNEL MODEL 29

III. NUMERICAL RESULTS AND DISCUSSION 33

A. PLOTS FOR MARGINAL AOA PDF IN AZIMUTH PLANE 33

B. PLOTS FOR MARGINAL AOA PDF IN ELEVATION PLANE 37

IV. CONCLUSIONS AND RECOMMENDATIONS 45

A. CONCLUSIONS 45

B. RECOMMENDATIONS 46

vii

LIST OF REFERENCES 47

INITIAL DISTRIBUTION LIST 49

Vlll

EXECUTIVE SUMMARY

This thesis provides closed forms which describe the statistics of the angular
distribution in azimuth and elevation planes for geometrically based single bounce spheroid
model. This spatial model can be generated by revolving an ellipse, which lies on the xz
plane of a 3D Cartesian coordinate system, about the x-axis. This ellipse has semi-major
axis a and semi-minor axis b, respectively. The base and the mobile stations are located in
the xy plane with the mobile station being at the center of the spheroid and the base station
at the origin of the coordinate system. It is assumed that the mobile station is transmitting
and the base station is receiving. The base and mobile stations are separated by a distance
D.

The scattering spheroid model is applicable to a macrocell environment in which the
received multipath signals at the base station are originated after a scattering from the
surrounding environment about the mobile station. Assuming that the scatterers are
uniformly distributed with a constant scatterer density function in the volume V as defined
by the geometry of the spheroid, we derive the marginal angle of arrival (AOA) probability
density functions (p.d.f)- in the azimuth and elevation plane.

Using the derived closed forms for both p.d.f.s we generate plots to examine how
the parameters a, b and D affect the angular distribution of the received waves. From the
marginal AOA p.d.f. in the azimuth plane we conclude that:

i\

1. The higher the ratio a/D, the lower the probability for the received
multipath signals at the base station to be confined to a small angular
region centered about the axis which connects the base and the mobile
stations.

2. The angular distribution in the azimuth plane is independent of the length
of the vertical dimension b of the spheroid .

3. Comparing the results of the spheroid model for small values of b with
those of the circular scattering model presented in [Ref. 3], it was
observed that there exists a significant difference between them.

Additionally, the effect of the a/D ratio on the r.m.s angle spread in azimuth plane is
discussed. From the plot results of the r.m.s angle spread in azimuth plane versus a/D,
we conclude that for values of a/D up to 0.5, there exists a linear relation. However, for
higher values of a/D, the relationship changes to a nonlinear curve with a positive
gradient.

Regarding the marginal AOA p.d.f. in the elevation plane, the following
conclusions were drawn from the plots:

1. Similar to the p.d.f in the azimuth plane, the higher the ratio b/D the lower
the probability for the received multipath components at the base station to
be restricted in a small angular spread around the axis, which connects the
base and the mobile stations.

2. The r.m.s angle spread which also depends on the semi major axis a, has
a linear dependence with respect to the b/D ratio.

AKNOWLEDGMENT

The author would like to thank Prof. R.Janaswamy for his support and his
guidance through every step of this thesis, my wife, Maria, for all her patience, and
understanding.

XI

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xn

I. INTRODUCTION

A. BACKGROUND

The outdoor radio environment contains a large number of obstacles, such as
buildings, cars, hills, etc, which obstruct, reflect and scatter the wireless radio signals. As
a result, the transmitted signal arrives at the mobile station or at the base station from
many different angles and times of arrival; this is known as multipath. Another common
phenomenon in cellular systems is the co-channel interference, and it is created because
in a certain geographic region(cluster) there are other subregions(cells) that utilize the
same set of frequencies. Multipath fading and co-channel interference can increase
significantly the bit error rate and they don't allow high bit rate data services in wireless
communications systems. The presence of multipath and the motion of the receiver or the
transmitter causes the received signal envelope to be time-varying and experience a
Doppler shift that depends upon the angle of arrival of the incoming waves. Concepts
such as angular spread, time delay spread, and Doppler spread have to be taken in
consideration during a wireless communication system evaluation. In particular, they
should be taken into consideration in the design of smart antennas used for diversity,
beamforming and emitter localization applications.

The multipath effect alone can be countered by applying antenna spatial diversity
which implements an antenna array consisting of multi-elements. Spatial diversity is an
arrangement whereby signals from multiple elements are combined to result in a signal
that fades less rapidly than the individual element signals. Ideally this can be

accomplished if received signals at the two elements are decorrelated. The degree of the
decorrelation between the two received signals by the two elements can be expressed by
the spatial cross-correlation coefficient which is a function of the joint p.d.f. f^e{<p,d)

describing the angular distribution of the incoming waves in the three-dimensional space
at the receiver.

The interference limited factor can not be overcome by using spatial diversity.
Switched beam arrays can be used with some success if the angle of arrival of the
interfering signal is not near the angle of arrival of the desired signal [Ref. 1]. Only the
beamforming arrays have the potential to completely suppress the co-channel interference
( if the number of elements exceeds the number of co-channel interferers) improving the
wireless communication system performance. The performance of these type of smart
antennas depends on the spatial characteristics of the wireless radio channel. The term
"spatial characteristics of the radio channel" is used to mean the statistics of the received
signal envelope in the spatial domain. These spatial characteristics may not always be
available from measurements. In such a case one needs to devise geometric models to
depict the scattering process in the spatial domain. Another area where knowledge of
angular distribution is valuable is in the determination of the power spectral density [Ref.
1]. Therefore the angular distribution of the received multipath components is important
in evaluating the performance of radiowave link between the transmitter and the receiver
in wireless communications.

Spatial channel models have been created and applied in order to be able to derive
the statistics of the received signal envelope for both macrocell and microcell
environments. The term macrocell enviroment assumes that the scatterers surrounding
the mobile station have the same height or are higher than the antenna of the mobile.
This results in a received signal at the mobile station which arrives from all directions
after bouncing from the surrounding scatterers. In contrast, the base station's antenna is
located higher than the surrounding scatterers. Therefore the received multipath signals
at the base station originate after scattering from the area near to the mobile station
[Ref.2]. In microcell enviroment both base and mobile station are deployed in the same
height as the surrounding scatterers. This implies that the scattering process takes place
near to the base and mobile station. The characteristic difference between the types of
environments is that the angular spread of the received signals at the base station is
confined to a small region for macrocell environments. In the microcell enviroment this
region tends to be larger.

In Table 1, which is a modified version of a table from [Ref 2.], are listed some
representative spatial channel models.

SPATIAL
MODEL

CHANNEL

DESCRIPTION OF THE
MODEL

APPLICATIONS

1. Lee's Model

Scatterers are evenly spaced
on a circular ring about the
mobile at a distance R. Each
of the scatterrers represents
the effect of many other
scatterers and hence is
referred to as an effective
scatterer. The base station is
deployed a distance D from
the mobile station.

Predicts correlation

coefficient using a discrete
angle of arrival(AOA)
model. A modified version
of this model can account
for a Doppler shift,
assuming that the scatterers
are moving with an angular
velocity.

2. Discrete Uniform
Distribution

A model similar to Lee's
model with the N scatterers
evenly spaced over an
AOA range.

Predicts correlation

coefficient using a discrete
AOA model. The

correlation predicted by this
model falls off more quickly
than the correlation in Lee's
model.

3. Geometrically
Circular Model

Based

In this model it is assumed
that the scatterers are
uniformly deployed within a
radius R about the mobile.
The locations of the
scatterers are defined by a
spatial scatterer density
function.

It is suitable for macrocell
environments and predicts
the joint time of
arrival(TOA) and AOA and
marginal density functions
at both base station and
mobile station.

4. Geommetrically
Elliptical Model

Based

Scatterers are uniformly
distributed in an ellipse
where the base station and
mobile station are the focii
of the ellipse.

It can be determined the
marginal AOA, TOA, the
joint TOA and AOA density
function. It is applied in
microcell enviroment.

5. Gaussian Wide Sense
Stationery Uncorrellated
Scattering (GWSSUS)

N scatterers are grouped
into clusters in space such
that the delay differences
within each cluster are not
resolvable within the
transmission signal BW.

Provides an analytical
model for the array
covariance matrix.

6. Gaussian Angle of
Arrival (GAA)

Special case of the
GWSSUS model with a
single cluster and angle of
arrival statistics assumed to
be Gaussian distributed
about some nominal angle.

Provides an analytical
model for the array
covariance matrix.

Table 1. Description of some representative spatial channel models. "From Ref. [2]."

Figure 1 through 4 demonstrate the first four spatial channel models, listed in
Table 1.

c/-HBase station

Figure 1. Lee's Model "From Ref. [2]."

Base station

"-,":"' //V e -':'

Eff^ctive^H
scattered

: M" :-

X

Figure 2. Discrete Uniform Distribution Model "From Ref. [2]".

Base
station

Scalterer region

Figure 3. Geommetrically Based Circular Model "From Ref.[2]".

Figure 4. Geommetrically Based Elliptical Model From Ref.[2]".

Every spatial channel listed in Table 1 has one common feature; the received
signal at the base station is assumed to arrive from a two dimensional plane connecting
the tips of the transmitter and the receiver. Hence the angle of arrival (AOA) includes
only the azimuth angle information.

In this thesis we consider a 3D spheroid model as shown in Figure 5 This
spheroid can be generated by revolving an ellipse, which lies on xz plane, with semi-
major axis and semi-minor axis a and b, respectively, about the x-axis. The base station
and the mobile station are located in the xy plane with the mobile station at the center of
the spheroid, and the base station at the origin. It is assumed that the mobile station is
transmitting and the base station is receiving. The distance between the two terminals is
D with the requirement that the semi-major axis of the spheroid be less than or equal to

the distance D. In the model considered here it is obvious that the angle of arrival (AOA)
of the received signals includes both the azimuth and elevation plane coordinates.

Figure 5. The spheroid geometry.

B. OBJECTIVE

Assuming that the scatterers are uniformly distributed around the mobile station
as defined by the geometry of the spheroid, the objective is to derive the marginal
probability density function (p.d.f.) angles of arrival (AOA) in the azimuth and in the
elevation planes at the base station. After extracting these quantities, we investigate how
the angular spread in azimuth and elevation plane are related with the dimensions of the
spheroid and the distance between the transmitter and the receiver, as well as how they
are correlated with each other.

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10

II. PROBLEM FORMULATION

A. INTRODUCTION

In this chapter we consider a single bounce scattering spheroid model. The model
can be used in a macrocell environment where the base station antenna height is
relatively high. As a result there will be no scattering from scatterers near the base
station. Figure 6, shows the spheroid scatterer density geometry and the notation used to
derive the marginal p.d.f.s. As is shown in the Figure 6, the location of each scatterer is
defined by the Cartesian coordinates^, y,z) or the spherical coordinates (rb,0b,6b), or

{rb,(f>b,0) , where ft represents the complementary angle of 0b in elevation plane.

11

Figure 6. The spheroid scatterer density geometry.
The method for the derivation of the joint TOA/AOA p.d.f. and marginal AOA p.d.f. for
both azimuth and elevation planes is based on the approach presented in [Ref. 3]. In the
Figure 6, it is assumed that scatterers are distributed randomly in the volume V of the
spheroid according to the spatial scatterer density function fxyz(x,y,z). It is further
assumed that the waves arrive directly at the base station after undergoing scattering only

12

once from within the volume V. Multiple scattering of waves is not considered in this
thesis.

Before proceeding with the determination of the p.d.f.s in the azimuth and the
elevation plane we would like to present the formulation for the joint TOA/AOA p.d.f
and marginal AOA p.d.f in the general case. In section B we start with the derivation of
the joint TOA/AOA p.d.f.. In section C, we show formulation for the marginal AOA
p.d.f.s. In the following sections D through G we derive the marginal AOA p.d.f in
azimuth and elevation plane for the spheroidal model.

B. DERIVATION OF THE JOINT TOA/AOA P.D.F.'S

Figure 7, shows a plane wave impinging on a base station after scattering from an

obstacle, represented by a point 'S' in space with the ordered rectangular triples (x, y,z).

13

S,(x,y,z)

Figure 7. The 3D used coordinate system.
The scatterer probability density function fXjytZ (x, y, z) can be also expressed

terms of spherical coordinates systems ( rb,(/)b,6b) as follows. The equations relating the
spherical coordinates to the Cartesian coordinates are:

-V?

1 9

+ y" +z"

(1)

6u = cos'

V*2+j

V"+Z"

(2)

14

^=tan"1(f)

(3)

x = rb sin 0b cos <pb
y = rbsin0bsm^b

Z = rbcos0b

(4)

(5)

(6)

where (x,y,z) and (rb,(f)b,9b) denote the location of the scatterer in Cartesian and
spherical coordinates respectively. The joint p.d.f fr + Q \fb,(pb,0b) can be found using

the well known transformation [Ref . 4] :

frb,<pb,eb(rb>0bA)=J(rb,</>b>0b)fx,yAx'y>z)

x = rb sin 6b cos <j)b
y = rb sin 0b sin </>b
Z = rb cos 6b

(7)

where j(rb,<pb,0b) is the determinant of the Jacobian transformation given by:

J(rb>fo>0b)= Q-Mr-ttr = rb s™0b> o<eb<7r

dx dx

dx

drb d<pb

deb

dy dy
drb d<pb

dv

deb

d^JZ-

dz

drb d(pb

d9b

(8)

Substituting the equation (8) into (7) we get for the scatterer probability density function
fr<p9 vb>$b>@b) w^ respect to spherical coordinates:

.2 •

Ail (rb,fb,Ob) = rb sin0bf (rb sin 6^ cos<pb,rb sinfy, sm</>b,rb cos0fe)

(9)

If we want to express the scatterer probability density function fr $ q Vb^b^b)
with respect to (T,0b,0b), where T denotes the time delay of the multipath component,
we have first to find the relationship between rb and the T. Applying the law of cosines
to the triangle BS,S,MS we get the following equation:

15

rf = rg + D2 - 2rbDcosS (10)

where the cos 8 is called the 'cosine direction' given by:

cos S = sin6b cos(f)b . (11)

The delay corresponding to the total path traveled by the wave is:

r = -^p-= £ \rb +4^ + D2 + 2rbDsinOb cos^ ) (12)

where 'c' is the speed of propagation of waves.

Squaring both sides of the equation (12) and solving for rb gives:

b 2{D sin 0b cos <pb-Tc) v }

The joint TOA/AOA p.d.f. / . Q \T,<pb,9b) can be found using the Jacobi
transformation [Ref. 4] according to:

JzA ,9b \T> <Pb#b)- \jMhA)\

an

rb = ,tn?l-t*L-~\ d4)

* 2(Dsm<9z?cos^-rc)

where \J\rb,(pb,6bJ is the determinant of the Jacobian tranformation given by:

\j(rb>fo>0b] =

drb

dr

2(Dsin^cos0z,-rc)" ,* c\

D2c+r2c3 -2tc3D sin 6b cos

Substituting the equation (15) into (14) we get for the joint TOA/AOA p.d.f.

f (T a. a ,_ D2c+T2c3-2rc2Dsmdbcos0b f ( p2-T2c2 A n) (ifft

Jr^eb Mb^b) ~ 2{Dsm9b.C0S<Pb_Tc)2 frbM Uds^cos^-zc)^^!- (">)

Using the last equation (16) and the equations (9) and (13) we can express the joint
TOA/AOA p.d.f. / ' 0{T,(pb,&b) in terms of original scatterer probability density

function fx v _(x, y,z) as follows:

16

(D2-r2c2)2(D2c+T2ci-2tc2Dsindbcos4>b)

r , n n • n ( L> -T C ) (U C+T C -4ZC U Sin tfh COS«>j, 1 . / . „ . . _ . „ \

fz,0b ,eb (?,foA) = sin eb rTT— " a fx.y.z Vb sm 0* cos $, , rb sin ^ sm (pb , rb cos ^ ) •

" o(i'sin&6 cos0j,-arj

(IV)
Note that the equation (17) expresses the joint TOA/AOA p.d.f. / Q [T,<pb,0b) as

observed at the base station in terms of an arbitary scatterer probability density function

/w (*•**) ■

When the scatterers are uniformly distributed within an arbitrarily space A with a
volume V, then the scatterer probability density function fx Z[X, y,Z) is given by:

fx,y,z(x>y>Z)=\ ■ (18)

[0, otherwise
If we consider this special case, the joint TOA/AOA p.d.f. / . Q [T,0b,0b) can be

written:

f (t <h G \-s.™f) (£>2-T2c2)2(D2c+T2c3-2TC2Dsm9bcos<t>b) f.Q.

Jr,<pb ,Bb yT><Pb'°b)- sm °b 01//n . ~ " S V l V)

Yb ° %V\Ds\Yi6bco^(t)b-rc)

where the range for the values of T, (f)b , 6b has to be defined.

C. DERIVATION OF MARGINAL AOA PROBABILITY DENSITY
FUNCTION

The marginal AOA p.d.f. with respect to spherical angle coordinates could be
found by integrating the joint p.d.f. fr , e fo,^»0$) , as given in equation (9) with

respect to rb over the range rb ypb,0b) to rb \<f>b,0b) , and then either with respect to

0b\<f>b) over the range 0b(<pb) to 0b\(t>b) or with respect </>b{0b) over the range

(f)b (0b ) to ^ \0b ) depending on whether the marginal A.O.A p.d.f. is to be found in

17

azimuth or elevation plane respectively. The terms rb(0b,9b) and rb (0b,9b) are the
lower and the upper limits of integration and depend on the azimuth angle <pb and the
elevation angle 6b . The values of these limits can be found from the intersections of a
straight line 6b = constant, (f)b = constant with the scatterer volume. The limits of
integration in 9b , dblm<S9b0for the marginal AOA p.d.f. in azimuth plane can be
defined as the smallest and the largest 9b -values that bound a region R, which is the
intersection between a plane at a specific 0b value and the scatterer region volume. The
above statement can be expressed in mathematical expressions as follows.
The marginal AOA p.d.f. f^ ((/>b) in azimuth plane is given by:

UMb)=i)A jfrb,M(rb,<t>b,0b)drbd0b =

fr] V^i I

r„Mb^bl

j* rb sin 9b fx,yyZ (rb sin 9b cos 0b , rb sin db sin (j>b , rb cos 9b ]drbdQb . (20)

The marginal AOA p.d.f. fe \0b ) in elevation plane is given by:
(A A)

fe»^ = lyb) \frbAA^rb^bA)drbd(l)b =

<Pb2(ebK(<Pi,A)

j rb2 sin 6b fxyz (rb sin Qb cos (j)b , rb sin 9b sin </>b , rb cos 9b )drb d<pb . (21)

The limits for (pb integration, which depend on the elevation angle 9b , can be
determined from the smallest and the largest azimuth angles that bound the region of
intersection between the cone 9b = constant and the scatterer volume.

18

Equations (20) and (21) express the AOA marginal p.d.f.'s for any scatterer
probability density function fx \x,y,z)- When the scatterers are uniformly

distributed in a volume with a constant probability density function fx z[x, y,z), the

marginal AOA p.d.f. f^ {(pb) is given by:

(22)
and the marginal AOA p.d.f. fe \6b ) is given by:

h2{eb)rb2{<Pb,eb) hzfo)

AW4 J jrb2sm0bdrbd0b=^ jl^Ah^A)^^^

(23)

D. MARGINAL AOA P.D.F. IN AZIMUTH PLANE FOR THE SPHEROID
SPATIAL CHANNEL MODEL

Before we proceed to find the marginal AOA p.d.f. in the azimuth plane, we have
to define the equation which describes the geometry of the spheroid in Cartesian and
spherical coordinates. The spheroid in Cartesian coordinates is given by:

i*z£fcl +4 = 1 (24)

a" b~

and in spherical coordinates is given by:

{a2 cos2 6 \ + b2 sin2 6b)r2 -2Db2rbsin6bcos0b + b2\D2 -a2) = 0 . (25)

19

Instead of the polar angle 6b , its complementary (3 = ^ — 0b will be used hereafter for
simplicity. Equation (25) can be expressed as:

A(/3)rb2 - B(f3)rb cos <j)b + C = 0 (26)

where, A(/3) = a2 sin2 (3 + b2 cos2 /3 , B(/3) = 2Db2 cosj3, C = b2(D2 - a2) and
A{j3) = B2 (P)cos20b- 4 A(p)C

= 4b2[a2(a2sin2 j3 + b2 cos2 j3)-D2a2 sin2 J3-D2b2 cos2 Psm2<pb}.
The two roots of the above equation are:

/ ps Db2 cos P cos <pb-^ a2 b2 [a2 sin2 0+b2 cos2 j3j-D2b2\a2 sin2 (3+b2 cos2 /?sin2 <j)b)

% {<Pb'P) = 1 2 ■ 2 a S 2~Ti

1 \az sinz p+bz cosz P)

/ ns Db2 cos Pcos<pb+Ja2b2\a2 sin2 P+b2 cos2 P)-D2b2[a2 sin2 P+b2 cos2 psin2 <pb

rbA<Pb'P) = S / 2 . 2 ' " 2 2,1

(azsinz/?+6zcosz/?j

(27)
In terms of the angle /? , the marginal AOA p.d.f . f^ (</>b ) in azimuth plane expressed

as:

/*(*») = £ jK(^,£)-r^,£)]cos£# (28)

20

D cos <j>,

(j)b = cons tan t

■J a2 - D2 sin2 <t>b

Figure 8. The intersected region R between a plane <j)b = constant
and the spheroid.

Figure 8 shows the limits of the angle /? for a specific value of (pb . The limits of

integration in the elevation angle can be determined from the values of the angle J3 that

bound the region R, which is produced by the intersection between the plane (f)b =constant

and the spheroid. In our case the cross section of the spheroid with a plane (j)b - constant

Vi "> i . .

«" - D" sin" <pb and semi-minor axis

equals to -«\a2 - D2 sin2 </>b and the center located at Dcos(j)b . For a general angle /?,

the straight line from the origin intersects the region R at two points r^ and r^ . At the

two boundary points the roots >i and ru of the quadratic equation (26) in rb coincide.

This happens when the discriminant of the quadratic equation is set to zero. Carrying this
out, the upper limit /?max is given by:

21

Anax = tan

-1

r

b \g--D -sirr^

a V D2_fl2

(29)

This equation is valid for <f>b in the range of sin (- -g-J < ^ < sin Hjj. These
correspond to the values of ^ that bound the spheroid in the xy-plane. Taking all of

the above into consideration, the marginal AOA p.d.f. in azimuth plane is given by:

Lift)

f*W>) = W Jcos^^^J-^^^^^in'l-fj^^^sm-1^). (30)

0

The evaluation of the integral as defined in the equation (30) is straightforward.
The following relations simplify some of the terms:

= (^%f^-^(^>^X^(A.^ + r44(A,^(^f^) + ig(^>^)). (31)

%{<i>b,P)-\(<Pb,P)=i!k

_ B(0)cos0h

rblWb,P)-rbi{4>b,P)=1fc

(32)
(33)
(34)
(35)

Using equations (31) through (35) in equation (30) we get:

Anax (& ) , .

f*,Wb) = W ]{% -^K + Vb, + rb2Jcos fid/3 =

o

AnaxWfc)

3V J

^B2(/3)cos2^-4A(j3)C Ib2(/3)cos2
MP) \ A2(/3)

C
A(£)

^cosfidfi

(36)

22

At a specific value of <f>b when P = Pm2X the discriminant becomes
A(A„ ) = 5: (^ ) cos2 ^ - 4A(#nax )C = 0 . Hence, wnting

= [£2 (/?)cos2 A - 4 A(^)C] - [£2 (/?max )cos2 fa - 4A(/?max )C] ,
we see that
A(^) = 4^2[a2(a2sin2^ + ^2cos2^)-D2a2sin2^-D2Z?2cos2^sin2^ -

-4b2[a2[a2 sin2 0max +b2 cos2 J3max)-D2a2 sin2 jflmax -Z)V cos2 y#max sin2 ^ j,

(37)
which gives:

A = 4Z72[sin2?mu-sin2^Ia2(D2-a2)+Z?2(a2-D2sin2^)]. (38)

Using this the integral in equation (36) can be written as follows:

r (± \ _ 160V cos2 0b^a-{Dl-a-yb2{a--Dz%m2$b) f cos3 /g-y/si^fl^-siir 0 ,g
J<pb\n)- W J (a-sin2/?+62cos:/?)3 aP

4(D2-a2)b^a2(D2-a2^b2(a2-D2sm2<pb) V cos^sin^-sin2/? ^

3^ J (a2sin2/?+*2cos2/?)2 P' K }

The first integral in equation (39) can be simplified as:

"max t. „ I~. 5 I 5 « "n

'max 3 _ I ■ 2 a 2 /> ^max _ . ? o, /~~ 2~o 2 a

r cos ^sin /?max -sin /? _ r cosy3(l-sin yg^sin /?max -sin yg
J (a2sin2y3+62cos2y5)3 J (a2sin2/?+62cos2/?)3

On using the substitution t = sin (3 , and fmax = sin /?max the last integral becomes:

"max _,, . i - f" •> _ '. 7 /» max /, -> \ /~t 7

J (ahm20+b2cos2/3)3 P~ J [^+(fl2_fe2^f

23

Hence the term 1— t can be written as follows:

i-<2=-^l-«24242-62)2)J.

Using this in equation (41) we get:

'max I ■> \ Hj 2 max H> 2 max /"^ 2*

f '-rNU-r,f = ^ [ V^CL -df — -L- f 3fa± <ft (42)

J N^W fl -& Jo N*2-*2W a * Jo N*2-»2W

Furthermore, writing t = / sin a , we get for the last two integrals:

T /I 2 ? 2

f j^-f^-^dt = Ij- ^^xcos-or »a (43)

» \b2+{a2-b2)t2) JQ (b2 +(a2 -b2 )t2 Tnzx sin2 af

and

'max r^ - 2

I . V?max~? =<ft = U f maxCOS a ^da (44)

J (6-+(a--fe-)/-f J(fe-+(fl--fe-)rmaxsin-af

Similarly, the second integral in equation (39) can be expressed as:

a S.

Pmax _ I . 2 a ■ 2 o tmSx (~7 2 % 2 2
J COS^Vsm A-x^° ^ = J Vrmax-r <ft = J ^^ ^ . (45)

q (fc2+(a2-£2)sin2/?)~ q (t2+(a2-fe2)r2j q (fe2+(a2-i>2)rmaxSin2Qrj

The last integrals described in the equations (43) (44) and (45) are evaluated using the
formulas 2.563-1, 2.563-2, and 2.563-3 from [Ref. 5].
The marginal p.d.f. in azimuth is then given by:

U(h)=lM-lM)- sin-'f-^ASsin-'fe) (46)

. / \ ;rD2cos2^Va2(D2-a2K*2(a2-D2sin2^) (2a2-4b2)y2+Lb4-2a2b2}y-a2b4 ,A~,

where lx [<f>b ) = "— J ^ '- J ' ^ - ' (47)

Y^y\a--b-j

and IM„)= ^--^Mo^V-o2™-^ {,-?). (48,

iV i]y{a--b-}

24

with y = a2 sin2 J3max +b2cos2j3T]

E. MARGINAL AOA P.D.F IN AZIMUTH PLANE FOR THE SPECIAL
CASE OF THE SEMI-MINOR AXIS OF THE SPHEROID GOING ZERO

In this section we examine the special case of marginal AOA p.d.f., given by
equation (46), for values of 6— »0. In order to evaluate the p.d.f. for this case the
following simplifications are made on equations (47) and (48):

y = a2 sin2 £raax +b2 cos2 ^max = b2 cos2 /?max (l + £ tan 2 /?max)

where tan /?max is given by equation (29). Since b —» 0 the last equation can be

^2 2

expressed as follows: Y -,cos, • (49)

D —a

1 2/2 2 1

The terms y-b and (y-b )\D -a J can be expressed as :

y-b2 =b2a2-D*sm;*b >0md (y-b2)(D2 -a2)=b2(a 2-Z)2sin2^ J. (50)

Plugging equations (47) and (48) to equation (46) the latter can be written as :
I U)-I U )- £ MP2^2M«2-P2*"2ft)

-(D2-a2)(r-^2)+^^[(3a2-4^2V2 + (4^-2flVV-«V

(51)

The term [3a2 -4b2 jy2 + \4b4 -2a2b jy-a2b4 — can be expressed as:
\3a2 -4b2)y2 + (4b4 -2a2b2)y-a2b4}± = 3a2y-2a2b2 -a2^-4b2y + 4b4. (52)

25

The first three terms in the right side of equation (52) are second order terms, while the
last two are fourth order terms with respect to b therefore can be neglected. Hence
equation (52) reduces to:

3a2y-2a2b2 -a2^--4b2y + 4b4 =r4^

[D'-a'

3D2 cos2

i°2-°2{-

D'-a-

\D2-a2\ + -£=*
D cos

-> \

(53)

Combining equations (49), (50) and (53) and taking also in consideration which terms are
second or fourth order with respect to 'b', equation (51) can be written as follows:

'lk)-'2(A)s#fe&

a2 +D2 sin2 <j)b + D2 cos2 (f)b

/ o 2

.9 -> ^

3D' cos' <pb ^ D^-g-
\D--a) D-cos"0b

Carrying out some simplifications in equation (54) we get:

4aJ

(54)

(55)

Finally the marginal AOA p.d.f in azimuth plane of the spheroid for the values of b — > 0

is given by:

UM) = jjj{"2-D2™20b) sin^-fj^^^sin-1^). (56)

F . DERIVATION OF MARGINAL AOA PDF IN AZIMUTH PLANE
THROUGH A SECOND APPROACH

As we have mentioned before the marginal AOA p.d.f in azimuth plane is given

by the:

f,hW = V J jrb2sm0drb0 = ±j \rbrb cos/3drbd/3.

(57)

26

D cos Qh - a cos {n-a)

Dcos(|>

ya2 - D2 sin2 <pb

Figure 9. The 2D coordinate system used for the derivation of the azimuth p.d.f.

Figure 9, shows the intersected region between a plane at a specific (f)b value and
the scatterer region volume. In the plane <pb ^constant, we adopt a 2D Cartesian
coordinate system Xiyi. The terms of the integral in the equation (57), rb cos j3 and
rbdrbd(3 can be written as:

rb cos fi-D cos (p + A cos a and rbdrbdj3 = d^dv, = dx2dy2

(58)

In the X2y2 2D Cartesian coordinate system as shown in Figure 9, the term A cos a = x2

27

According to the above equation (58) equation (57) can be written as:

Anax rb2(h-P) ^a2-D2 sm2<pb yla2-D2sm2<pb

ffa (fo) = y jrb cos PrbdrdP = f j(Dcos</>b+A cos a]dx2dy2

-AnaxM^) ° °

(59)
The integral in equation (59) with respect to polar coordinate system A and a can be
expressed as:

f0b (0*) = v J J (D cos 0b + A cos otydMa ■ (60)

ar=0 0

Taking in consideration that the term A cos a is equal to X2, since X2 is an odd function,
equation (60) can be written as:

/*to) = v J J(0cosft,+jc2)fctate = £ jJDcos0bAdAda (61)

a=0 0 ^(^)

where A0 (^ ) is the area of region R.
Equation (61) is simplified as:

f^0b) = iDcos0b jJAdAda = ^A^(0b) (62)

A(&)

where A^ ((/>b) = 7T — [a -D sin"^j. Substituting the above term in equation (62) we

get for the marginal AOA p.d.f. in azimuth plane:

4W = ^M«2-fl2sm2<0 sm-(-f)<h<s,n-%). (63)

The last solution approach shows that the azimuth angular distribution is independent of
b and coincides with equation (56). Hence, we conclude that the limiting equation (56),
valid for b — > 0 , is also true for all b.

2.X

Finally, using the closed form expression for the p.d.f. we evaluate the r.m.s
angular spread in closed form:

<t>rms = 2<Pstd (64)

where

sin""

& = jriUM^h = (sin-'fef +f J^lsm-'^-^-f + |4. (65)

G. MARGINAL AOA P.D.F. IN ELEVATION PLANE FOR THE SPHEROID
SPATIAL CHANNEL MODEL

As noted in section C of the present chapter, the marginal AOA p.d.f. in elevation
plane is given by:

ffiiP) = V" J \ rl cos Pd<l>b=ir J k fe , /?) - i (<j>b , /?)]cos 0d*b (66)

^ (P)=fnax

ff(P)=& fK(<h>0)-rfa>0)]p*&*>. \fe0u <67>

0

The maximum elevation angle /?M is obtained from the values that bound the
spheroid in the xz-plane and can be found analytically from the equation (29) by setting
the azimuth angle Qb equals to zero:

■_1' h ^ (68)

/?M=tan-'(74=T).

29

Using the equations (31) through (35), the integral in equation (67) can be simplified as
follows:

fP(P)= w J* cos /% - \ K + \ \ + rl

Vmax I — / \

2cos/? f tJB- {p)cos-0b-4 A(/3)C (b2(/3)cos2 0b g_ U.

' 3V J A(0) \ AriP) A(fi)rrb
0

- I^f£ j WR^IC* • (69)

0

Substituting the functional forms of A{(3) , B{(3) and C from equation (26) we get:

2 4 3/5 "max i

ff{p)=vlvZl°?\tfV J ™?h4B2{PWh-*KP)cd<i>b-

H 3(a sin p+» cos p) V J

-^rfe^v / M^°s2fe-4A(/?)o% . (70)

3(<3 sin B+b cos p) V J
0

The upper limit for the azimuth angle 0max , for a fixed {3 , is the largest value of angle
0£ that bound the region R , which is generated by the intersection of the cone
j3 = constant with the spheroid. In general, the intersection will include all the points
ru<rb< rb^ , where ru and rb^ are given in equation (27). Once again, at the boundary
points it yields rb = rb-i . Thus 0max is obtained by setting the discriminant
A(<pb) = B2(/3)cos2 <pb -4A(/3)C equals to zero and solving with respect to^ :

^ax=sin-'tVl-(lS7^)2l (71)

30

In order to be able to solve the integral in equation (70) we have to simplify the
expression for the discriminant A(<pb) = Z?2(/?)cos2 (j>b -4A(/3)C under the square root.
This is done applying the same approach as in previous section D. The discriminant, at a
specific value of (3 becomes equals to zero when the azimuth angle (j)b coincides with

the maximum value of <j)b - 0max . Thus writing the discriminant as
Afo) = Afo)-A(0

= [B2 (/?)cos2 </>b - 4A(/3)C] - [B2(/3)cos2 0max - 4A(j3)C] , becomes
A((/>b)= \*D Vcos2 ficos2 <pb-\b2[p2-a2\a2 sin2 /3 + b2 cos2 /?)]-

- 4Z)Vcos2 /?cos2 0max -4b2{D2 -a2\a2 sin2 j3 + b2 cos2 /?)] (72)

which gives:

A(^) = A(^)-A(^max) = 4Z)Vcos2Asin2^max-sin2^). (73)

Substituting equation (73) into equation (70) gives:

3 6 4 o ^max /

~ ^ 2 ■ ^.2 ' 2 ^ I #m ^max-sin hd(t>b- (74>

3(a sin p+b cos p) V J

The above two integrals may be rewritten as:

0max

J

0

0max

f COS 2 0fc V(sin 2 ^max ~ sin 2 ^ )<% = sin ^max / cos 2 h JO ~2T sin 2 ^ Wi ^5)

_ . V Sln ^max

rraax , -rmax i

and <J(sir\ 2 0max - sin 2 </>b )d<pb = sin 0max /(l — -4- — sin 2 </>b )<% . (76)

_ _ V Sln ^max

These integrals can be evaluated in terms of complete elliptical integrals first and second

31

kind. Indeed with the help of the formulas 2.595-1 and 2.592-2 of [Ref. 5], these
integrals take the form:

Tmax

in0 cos"^.

sin

and

max

^(sm<d__.. )

(77)

sin^ J" ^(i___L_sin^^ = E(sin ^ ) - (sin^ - l>C(sin ^max ) (78)

max 0

where AT(sin 0^ ) and £(sin 0max ) denote the complete elliptical integral of the first and

second kind respectively [Ref. 5].

Substituting the results from the equations (77) and (78) into equation (74) we get
the final expression for the marginal AOA p.d.f., in elevation fn\P) in a closed form as

follows:

fp(P) =

3L6 4

16D-Vcos4/?

6(a2sin2/?+62cos2/?)3V

1 + sin 2 0max )£(sin 0max ) - cos2 0max K(sin 0max )

4Dfc4(p2-q2)cos2^
3(a2 sin2 P+b2 cos2 /3)2V

£:(sin^max)-cos2^max^(sin^max)J' \0\ ^ &M (79)

^D2-a~

with fiM = tan"1 T^-? , sin^ =ffi-(JJ^f and V =f ;za2£.

32

III. NUMERICAL RESULTS AND DISCUSSION

Having derived closed form expressions for the marginal AOA p.d.f. in azimuth
and elevation planes, we now present in this chapter some numerical results for each
marginal AOA p.d.f.. Recall that D is the distance between transmitter and receiver, a is
the semi-major axis of the spheroid, and b is the semi-minor-axis of the spheroid.

A. PLOTS FOR MARGINAL AOA PDF IN AZIMUTH PLANE

In this section we consider the cases of 'D', 'a' and 'b' as given in Table 2 and

generate some plots:

D(m)

a(m)

b(m)

Corresponding
Figure

1000

100

0.001

10

1000

100

50

11

1000

200

150

12

Table 2. Parameters of D, a and b used in plots for marginal AOA p.d.f. in the azimuth
plane.

Figure 10 has been plotted in order to check how the azimuth angular distributions

of the spheroid and the circular scattering models are related to each other. The circular

scattering model is presented in [Ref. 3]. The angular distribution for the circular model

is given by equation (80):

33

f&M

•3Dcos»VoW»-D2V,sin-l(-t)<fe<sin--y
0, else

(80)

S4

pdf of the spheroid for

D=1000m
a=100m

! / s"
1 ' t y

f^*-- \

b=0.001m

pdf of the circular for

D=1000m
a=100m

r - - t

!__ _ ./ J _ _ _

■J I

V l

— — T _ _ jj
1 ■■/'

i i i

1 1

-15 -10 -5 0 5 10

Angle of Arrival in Azimuth Plane(degrees),<|>b

15

Figure 10. Plot of f^ [tfo) versus^ forD=1000m, a=100m and b=0.001m. Also shows
the AOA p.d.f. for a circular spatial model for D=1000m and a=100m.

In Figure 10, we plot the p.d.f. for a/D=0.l and b/D=W6. Comparing the plot of the
spheroid model for small values of 'b' with this of the circular model, we can observe that
there exists a significant difference between them.

In Figures 1 1 and 12 we plot the p.d.f. for a/D=0.l, £=50m and for a/D=0.2,
6= 150m using the two different formulas for the angular distribution in the azimuth
plane.

34

~*4

pdf (eq(46) for

a/D=0.1 b=50m

— •— pdf eq(56) for
a/D=0.1

i _ _ i. i _ +_

jf j. ~v

f: i |

-15 -10 -5 0 5 10

Angle of Arrival in Azimuth Plane(degrees),<t>b

15

Figure 11. Plot of /,* (<pb) versus^ for D=1000m, a=100m and b=50m
(using the two different formulas).

4
3.5

3
2.5

r*2

1.5

0.5

pdf (eq(46) for

a/D=0.2b=150m

— t— pdfeq(56)for
a/D=0.2

/ \

1

_ L £ L 1 J A. _ J _ _ _

r j: r T 1 "V-|

-15 -10 -5 0 5 10

Angle of Arrival in Azimuth Plane(degrees),<|>b

15

Figure 12. Plot of /^(fo) versus <f>b for D=1000m, a=200m and b=150m
(using the two different formulas).

35

As shown Figure 1 1 and 12, the two different formulas give identical results for the
angular distribution in azimuth plane. Additionally, we conclude that the higher the ratio
a/D, the lower the probability for the received signals at the base station to be restricted in
a small angular region centered about the axis which connects the base and the mobile
stations.

Finally in Figure 13 we plot the r.m.s angle spread in azimuth plane as given in
equation (64) versus the ratio a/D.

60

50 -

40

■S 30

20

10

I I

i i i i i

: : ;

r.m.s angular spread 0

rms

M M M l/i

\ \ ! j/i \ ! I

I i^n } ': i i ':

Jn III!!!]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a/D

Figure 13. Plot of the r.m.s angular (jw spread versus a/D.

Figure 13 shows that for values up to 0.5, of the ratio a/D, the profile of the r.m.s angle
spread is almost linear. Note that the r.m.s angle spread in azimuth plane is independent
of b as proved with the derivation of the second formula for the p.d.f in azimuth plane.

36

B. PLOTS FOR MARGINAL AOA PDF IN ELEVATION PLANE

D(m)

a(m)

b(m)

Corresponding
Figure

1000

100

1

14

1000

100

30

15

1000

100

50

16

1000

200

50

17

1000

500

50

18

Table 3. Parameters D, a, and b used in plots for marginal AOA p.d.f. in the
elevation plane.

Figure 14 is generated to demonstrate the angular distribution in elevation plane
for small values of b.

37

800

700

600

500

S 400

300

200

100

I

] I

D=1000m

a=100m
b=1m

I

I
]
I
I

I

I I I

4-3-2-101234
Angle of Arrival in Elevation Plane(degrees),fS

Figure 14. Plot of fp{0) versus ft forD=1000m, a=100m
and b=lm..

As we can see form the Figure 14 the angular distribution in elevation plane for small
values of b looks like a dirac function as it is expected.

Figure 15 and 16 are plotted for demonstration purposes, giving information of
how the changes of the ratio b/D for a given 'a' affect the angular distribution in elevation
plane.

38

30

25

20

S 15

10

1 1 1 1 1

D= 1000m

a=100m
b=30m

r -i
i 1

;

i 1 7 ! i ! \

■

-3-2-1 0 1 2

Angle of Arrival in Elevation Plane(degrees),P

Figure 15. Plot of fp(0) versus j3 forD=1000m, a=100m
andb=30m..

Figure 15 shows the the angular distribution in elevation plane for a ratio b/D=0.03 and

a/D=0.l.

16

10

S 8

II ! I | .... .

i

D= 1000m

a=100m
b=50m

I I

L L

:-----i/---r---H----;----\

1

L /_J J J i

V

:/ i i i i

\:

-3-2-1 0 1 2

Angle of Arrival in Elevation Plane(degrees),f3

Figure 16. Plot offp(0) versus y# forD=1000m, a=100m
and b=50m.

39

Figure 16 shows the p.d.f for a ratio b/D=0.05 and a/D=0.1 As seen, compared to Figures
15 and 16, the higher the ratio b/D the lower the probability to receive multipath
components from a small region /3BW at the base station.

Figure 17 is generated, as set with Figure 16, in order to show the effect of the
length of the semi- major axis a on the angular distribution.

S 8

1 1 1 1 1

i

!____! I-/ _l _ AJ

— D= 1000m
a=200m
b=50m

•4"|n"4"-i\

\ t \ :

7

1 - T

7

/

-3 -2 -1-0 1 2

Angle ol Arrival in Elevation Plane(degrees),fS

Figure 17. Plot of ffi(0) versus j3 forD=1000m, a=200m
and b=50in meters.

Figure 17 shows the p.d.f for a ratio b/D=0.05 and a/D=0.2. Comparing Figures 16 and
17, it is seen that p.d.f in elevation plane is almost independent of the length of semi-
major axis of the spheroid for small values of the ratio a/D. Figure 18, is generated to
show the effect of higher values of the a/D ratio on the angular distribution.

40

14

12

10

S 8

4- 1 1- J + V -A

D=1000m

a=500m
b=50m

: :/ 1 : : \: :

: / \

t ■*/ 1 i t ~\ \\~ r

\. 1 1 1 + -( 1 V(_

-3-2-1012

Angle of Arrival in Elevation Plane(degrees),P

Figure 18. Plot of ffi{fl) versus j3 for D=1000m, a=500m,
and b=50m.

Comparing Figure 17 and 18, it was observed that there exists a difference between the

two angular distributions.

Finally in Figure 19 we plot the r.m.s angular spread of the angle of arrival in

elevation plane, as given by the equation (81), versus the ratio b/D for specific values of

a/D.

firm=2\]fi2ffi(fi)dfi

(81)

The mean value of the angular distribution at the base station is zero, and the integral is
evaluated numerically.

41

30

25

_20
w

CD

B 15

0Q.

10

I I

+

B fora/D=0.1

r rms

3 fora/D=0.2

rms

3_0fora/D=0.5

rms

—

0

0

1 0

2

0.3 0.4 0.5

b/D

Figure 19. Plot of the r.m.s angular (3,-™ spread versus b/D
for various ratios a/D,for the spheroid scattering model.

As shown in Figure 19, there exists a linear relation between the r.m.s angular spread and
the ratio b/D independently of the ratio a/D.

Eventually for validation purposes of the two derived, in this thesis, closed
forms of the angular distributions, we plot in Figure 20 both p.d.f. with a almost equals to
b (b is not made exactly equal to a to avoid dealing with limiting process).

42

A, , ,-.

1 1 I 1

a=100m b=99.999m

pdf in elev for D= 1000m

a=100m b=100m

' __ _.:x^

! A \ ,

_ J _/ 1 _ i 1

: / : : : \
:__/.:_ J l L__\_J

..-X.J--i--i-4--

..-/-J--i.-.i-X„,

.-./i...J..-.L-.i....AH

/ : ; : : : \

/ i i i : : \

Figure 20. Plot of f* (<pb) versus^ and fp{0) versus (3 for D= 1000m,
a=100m and b=99.99m and for D=1000m, a=100m and b=100m respectively.

Figure 20, shows the angular distribution in azimuth and elevation plane for
a/D=b/D=0.l. From the plot we can see identical results for both angular distributions, as
expected due to the symmetry of the model.

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44

IV. CONCLUSIONS AND RECOMMENDATIONS

A. CONCLUSIONS

In this thesis we proposed a geometrically based single bounce scattering 3D
model. It has been assumed that the scatterers are distributed about the mobile with a
constant density function in a volume as defined by the geometry of the spheroid. The
distance between the base station and the mobile station is D. The focus of the effort of
this research was in the derivation of the mathematical expressions which described the
spatial statistics properties of the received signals in the presence of the three-dimensional
multipath scattering.

After presenting a general approach for the derivation of the joint TOA/AOA and
marginal AOA p.d.f.s in terms of an arbitary scatterer probability density function, we
presented the derivation of the angular distribution in azimuth and elevation plane for the
spheroidal model in closed expression forms.

Making use of these quantities we studied the behavior of the angular spread for
each p.d.f as function either of the spheroid dimensions or of the distance between the
transmitter and the receiver. For either angular distribution the model results in a higher
probability of arrival along the line of sight direction. Also, we observed that the higher
the ratio of each spheroid's dimension over the distance D the lower the probability that
the multipath components are confined to a small region. An excellent agreement exists
in the plots which are generated from the two different formulas of the p.d.f in azimuth
plane.

45

Analytical and numerical calculations were performed for the r.m.s angular spread
in azimuth and elevation planes respectively. The effect of the ratio a/D or b/D on the
corresponding angular spread was investigated.

B. RECOMMENDATIONS

Further research is required to enhance this model. It will be a interesting to derive

the marginal time of arrival p.d.f or to investigate the effect of the angular spread on the
correlation observed between the received signals an antenna array. Another area of
interest should be the validation of this model. This could be done comparing with
measurements taken over similar macrocell environments.

46

LIST OF REFERENCES

1 . Janaswamy, R., Radiowave Propagation and Smart Antennas for Wireless
Communications, Kluwer Academic Press, 2000.

2. Ertel, R.B., Cardieri, P., Sowerby, K.W., Rappaport, T.S., and Reed, J.H.,
"Overwiew of Spatial Channel Models for Antenna Array Communication
Systems," IEEE Personal Communications., pp. 10-22, February 1998.

3. Ertel, R.B. and Reed, J.H., "Angle and Time of Arrival Statistics for Circular and
Elliptical Scattering Models, "IEEE Journal on Selected Areas in
Communications., vol. 17(11), pp. 1829-1840, November 1999.

4. Papoulis,A, Probability, Random Variables, and Stochastic Processes, 3rd ed.,
McCraw-Hill Book Co., 1991.

5. Gradshteyn, I.S. and Ryzhik, I.M., Table of Integrals, Series, and Products,
5th ed., Academic Press, 1994.

6. Liberti, J.C., and Rappaport, T.S., Smart Antennas for Wireless Communications,
1st ed., Prentice Hall, Inc, 1999.

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48

INITIAL DISTRIBUTION LIST

Defense Technical Information Center ...
8725 John J. Kingman Road, Suite 0944
Ft. Bel voir, VA 22060-6218

2. Dudley Knox Library

Naval Postgraduate School

411 Dyer Road
Monterey, CA 93943-5101

3. Chairman, Code EC

Department of Electrical and Computer Engineering
Naval Postgraduate School

Monterey, C A 93943-5121

4. Professor Ramakrishna Janaswamy, Code EC/Js

Department of Electrical and Computer Engineering
Naval Postgraduate School

Monterey, C A 93943-5121

5. Professor Tri Ha, Code EC/Ha

Department of Electrical and Computer Engineering
Naval Postgraduate School

Monterey, CA 93943-5121

6. LT. Christos Sasiakos

13 Dabaki Street

11526 E.Stavros
Athens, Greece

49

b/D2 22527-200 mj